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Trend Growth and Robust Monetary Policy

  • Kohei Hasui EMAIL logo
Published/Copyright: November 23, 2020
The B.E. Journal of Macroeconomics
From the journal The B.E. Journal of Macroeconomics Volume 21 Issue 2

Abstract

Recent monetary policy studies have shown that the trend productivity growth has non-trivial implications for monetary policy. This paper investigates how trend growth alters the effect of model uncertainty on macroeconomic fluctuations by introducing a robust control problem. We show that an increase in trend growth reduces the effect of the central bank’s model uncertainty and, hence, mitigates the large macroeconomic fluctuations. Moreover, the increase in trend growth contributes to bringing the economy into determinacy regions even if larger model uncertainty exists. These results indicate that trend growth contributes to stabilizing the economy in terms of both variance and determinacy when model uncertainty exists.

Keywords: model uncertainty; robust monetary policy; trend growth
JEL Classification: E50; E52
Corresponding author: Kohei Hasui, Faculty of Economics, Matsuyama University, Matsuyama, Japan, E-mail:

Acknowledgment

The author thanks Evi Pappa (the associate editor) and two anonymous referees for their valuable suggestions and comments. The author acknowledges financial support from JSPS KAKENHI Grant Number 17K13768.

Appendix

A.1 Deviation from the Approximating Model

Although the central bank designed its policy supposing the worst-case, it should be considered that no model misspecification exists. This is called an “approximating model”(Giordani and Söderlind 2004). The approximating model can be obtained by substituting the policy function of the nominal interest rate under robust policy i t = c i ( θ , γ ) u t + r t n into the undistorted model in Eqs. (2) and (3).[26] The policy function of the inflation rate under the approximating model is given as follows:

(A.1) π t a = c π a u t ,   c π a = 1 − ϵ / ( θ κ ( γ ) ) χ ( θ , γ ) ,

where π t a denotes the inflation rate under the approximating model. The rate at which the robust policy deviates from the approximating model in inflation is given as follows:

π t r − π t a π t a = c π − c π a c π a = ϵ θ κ ( γ ) − ϵ ,

where π t r denotes the inflation rate under the robust policy. We analyze how trend growth and the model uncertainty affect this deviation rate by differentiating θ and γ as follows.

(A.2) − ∂ 2 ( c π / c π a − 1 ) ∂ θ ∂ γ = − ϵ [ θ κ ( γ ) + ϵ ] [ θ κ ( γ ) − ϵ ] 3 ∂ κ ( γ ) ∂ γ .

The sign of (A.2) is ambiguous. When θ κ ( γ ) < ϵ and σ > 1, the sign of (A.2) is positive. In this case, the cost of the robust policy in terms of the inflation rate increases as the trend growth rises.

A.2 Calculation: Detection Error Probability

In this section, we describe in detail the calculation of the detection error probability. The overall definition of detection error probability is expressed as follows (Giordani and Söderlind 2004):

p ( θ ) = 1 2 × Prob ( L A > L W | W ) + 1 2 × Prob ( L W > L A | A ) ,

where L A and L W denote the values of the likelihood of the approximating model and worst-case scenario, respectively. The notations A and W denote the approximating model and worst-case scenario, respectively. Given the data from the model, the probability is calculated as the rate of wrong choices between the worst-case scenario and the approximating model. The detection error probability thus indicates the difficulty of distinguishing between models with and without misspecification. Following Giordani and Söderlind (2004, p.2376), we set θ so that the detection error probability was 20%.

To obtain the detection error probability, we generate a data of misspecification term for the worst-case and the approximating model (we express these as ν t w and ν t a , respectively) for sufficiently long periods T.[27]

Then, we calculate the relative likelihood r w and r a as follows:[28]

(A.3) r w = 1 T ∑ t = 0 T − 1 ( 1 2 ( ν t w ) ′ ν t w + ( ν t w ) ′ u t ) ,

(A.4) r a = 1 T ∑ t = 0 T − 1 ( 1 2 ( ν t a ) ′ ν t a − ( ν t a ) ′ u t ) .

Finally, we obtain the detection error probability as follows:

(A.5) p ( θ ) = 1 2 [ freq ( r w ≤ 0 ) + freq ( r a ≤ 0 ) ] .

A.3 Brief Description of the Model of F. Mattesini and S. Nisticò (2010)

The households’ problem is given as follows:

(A.6) max C t , N t ( j ) E t ∑ t = 0 ∞ β t C t 1 − σ 1 − σ ( 1 − N t ( j ) ) 1 − ζ , s . t . P t C t + E t [ ℱ t , t + 1 B t + 1 ] ≤ W t ( j ) N t ( j ) + P R t + B t − T t ,

where C t , N t , W t , ℱ t , t + 1 , B t , PR t , and T t denotes consumption, labor supply, nominal wage, SDF, state contingent claim, nominal profit from firm sector, and lump-sum tax, respectively. j ∈ [ 0 , 1 ] denotes the index of continuum households.

In Mattesini and Nisticò’s (2010) model, the employment agency packs the differentiated labor supplies from household N t ( j ) . By cost minimization in the employment agencies, the demand for differentiated labor is derived as follows:

(A.7) N t ( j ) = ( W t ( j ) W t ) − η t N t ,

where η t denotes the elasticity of substitution between different types of labors.

Firms’ price setting problem is given as follows:

(A.8) max P t r ( i ) E t ∑ k = 0 ∞ α k ℱ t , t + k [ P t r ( i ) − P t + k M C ] Y t + k ( i ) s . t Y t + k ( i ) = ( P t r ( i ) / P t ) − ϵ Y t + k .

where Y t , P t r , P t , and MC t denotes the output, price to be set optimally, aggregate price, and real marginal cost, respectively. i ∈ [ 0,1 ] denotes the index of monopolistically competitive firms.

The firm i’s production function and real marginal cost are given as follows

(A.9) Y t ( i ) = γ t A t N t ( i ) ,

(A.10) M C t = ( 1 − τ ) W t γ t A t P t

A.3.1 The Stationary Equilibrium

To obtain a stationary equilibrium, we divide consumption, output, and government spending (G t ) by γ t , and we express these de-treneded variables with “hat” as for the arbitrary variables X ˆ t = X t / γ t .

(A.11) E t [ ℱ t , t + 1 ] C ˆ t − σ ( 1 − N t ) 1 − ζ = β γ − σ E t [ P t P t + 1 C ˆ t + 1 − σ ( 1 − N t + 1 ) 1 − ζ ] ,

(A.12) E t ∑ k = 0 ∞ ( α γ ) k ℱ t , t + k [ P t r ( i ) − ϵ ϵ − 1 P t + k M C ] Y ˆ t ( i ) = 0 ,

(A.13) Y ˆ t D t = A t N t ,

(A.14) Y ˆ t = C ˆ t + G ˆ t ,

(A.15) M C t = 1 − ζ 1 − σ ( 1 + μ t W ) ( 1 − τ ) A t C ˆ t 1 − N t ,

where μ t W = 1 / ( η t − 1 ) denotes the wage mark-up shock. D t denotes the price dispersion.

A.3.2 Log-Linearized Model

Log-linearizing around the steady-state, the model system is expressed as follows:

(A.16) y ˆ t = E t y ˆ t + 1 − σ n − 1 ( i t − E t π t + 1 − ρ ) + σ n − 1 φ ( 1 − ζ ) E t Δ a t + 1 − σ n − 1 σ s c E t Δ g t + 1 .

(A.17) π t = β γ 1 − σ E t π t + 1 + κ ( γ ) m c t ,

(A.18) mc t = ( s c + φ ) y ˆ t − s c g t − ( 1 + φ ) a t + e t ,

(A.19) y ‾ ˆ t = 1 + φ s c + φ a t + s c s c + φ g t .

where π t , i t , , y ˆ t , and y ‾ ˆ t denote the log-linearized inflation rate, nominal interest rate, de-trended output, and de-trended potential output. We define g t = G ˆ / Y ˆ g ˆ t , e t = μ t W − μ W , ρ = σ ln γ − ln β , and s c = Y ˆ / C ˆ . Combining these equations, we obtain the following three-equation-style expression:

(A.20) x t = E t x t + 1 − σ n − 1 ( i t − E t π t + 1 − r t n ) ,

(A.21) π t = β γ 1 − σ E t π t + 1 + κ ξ x t + u t ,

where x t = y ˆ t − y ‾ ˆ t (output gap) and u t = κ e t .[29] r t n denotes the natural interest rate. r t n is given as follows:

r t n = ( ζ − 1 ) ( s c − 1 ) φ + σ s c ( 1 + φ ) s c + φ E t Δ a t + 1 − s c ( ζ − 1 ) φ + σ s c φ s c + φ E t Δ g t + 1 .

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Received: 2020-06-17
Accepted: 2020-11-01
Published Online: 2020-11-23

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Contributions
  3. Occupational Choice and Investments in Human Capital in Informal Economies
  4. Asymmetric Effects of Monetary Policy
  5. Trend Growth and Robust Monetary Policy
  6. Delegating Optimal Monetary Policy Inertia in a Small-Open Economy
  7. Dating Structural Changes in UK Monetary Policy
  8. International Historical Evidence on Money Growth and Inflation: The Role of High Inflation Episodes
  9. Inequality, Growth, and Congestion Externalities
  10. The Neoclassical Growth Model and the Labor Share Decline
  11. Environmental Taxes and Economic Growth with Multiple Growth Engines
  12. Macrodynamic Modeling of Innovation Equilibria and Traps
  13. Advances
  14. Unraveling News: Reconciling Conflicting Evidence
  15. Did the FED React to Asset Price Bubbles?
  16. Agricultural Trade and Structural Change: Evidence from Paraguay
https://doi.org/10.1515/bejm-2020-0133
Keywords for this article
model uncertainty; robust monetary policy; trend growth

Articles in the same Issue

  1. Frontmatter
  2. Contributions
  3. Occupational Choice and Investments in Human Capital in Informal Economies
  4. Asymmetric Effects of Monetary Policy
  5. Trend Growth and Robust Monetary Policy
  6. Delegating Optimal Monetary Policy Inertia in a Small-Open Economy
  7. Dating Structural Changes in UK Monetary Policy
  8. International Historical Evidence on Money Growth and Inflation: The Role of High Inflation Episodes
  9. Inequality, Growth, and Congestion Externalities
  10. The Neoclassical Growth Model and the Labor Share Decline
  11. Environmental Taxes and Economic Growth with Multiple Growth Engines
  12. Macrodynamic Modeling of Innovation Equilibria and Traps
  13. Advances
  14. Unraveling News: Reconciling Conflicting Evidence
  15. Did the FED React to Asset Price Bubbles?
  16. Agricultural Trade and Structural Change: Evidence from Paraguay
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